P o l a r C o d e s o v e r q - a r y A l p h - - PowerPoint PPT Presentation

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C a p a c i t y

Ameya Velingker

Carnegie Mellon University

Based on joint work with Venkatesan Guruswami SpCodingSchool – Campinas, Brazil (January 19-30, 2015)

Channel Capacity

Theorem (Shannon's Noisy-Channel Coding Theorem, 1948). For any discrete memoryless channel W, there exists an associated nonnegatjve constant I(W), known as the channel capacity such that: For any ε > 0, one can communicate at asymptotjc rate R = I(W) - ε with vanishing probability of miscommunicatjon. For any R > I(W), it is impossible to communicate at a rate R without non-negligible probability of miscommunicatjon.

Challenge: Construct explicit coding schemes (with effjcient encoding and decoding procedures) that obtain arbitrarily small gap to capacity! For rate R = I(W) – ε, the gap to capacity is ε.

Binary Polar Codes

• Polar codes give the fjrst known constructjon to achieve

capacity with the aforementjoned guarantees

• Rate approaches I(W) as N

⇾ ∞

• O(N log N) complexity
• For polar codes over binary alphabet, [Guruswami-Xia '13] and

[Hassani-Alishahi-Urbanke '13] independently show the speed

• f convergence to capacity

– There is an absolute constant c such that any block length N ≥

(1/ε)c will yield a rate of at least I(W) – ε

Questjon: What about polar codes over non-binary alphabets?

Main Result

• We extend the work of [Guruswami-Xia '13] to q-ary polar

codes for any q > 2

• q-ary polar codes are the fjrst explicit deterministjc

constructjon of q-ary codes approaching the capacity of q- ary symmetric DMCs with provable guarantees

– Encoding and decoding complexity polynomial in block

length N

– Constant c = c(q) such that rate ≥ I(W) - ε for N ≥ (1/ε)c – Decoding error probability exp(-N0.49)